Linear representation theory of semidihedral group:SD16

Summary

We shall use the semidihedral group of order 16 with the following presentation:

$\langle a,x \mid a^8 = x^2 = e, xax^{-1} = a^3 \rangle$ .

Item	Value
degrees of irreducible representations over a splitting field (e.g., $\mathbb{C}$ or $\overline{\mathbb{Q}}$ )	1,1,1,1,2,2,2 (1 occurs 4 times, 2 occurs 3 times) maximum: 2, lcm: 2, number: 7, sum of squares: 16
Schur index values of irreducible representations over a splitting field	1,1,1,1,1,1,1
smallest ring of realization (characteristic zero)	$\mathbb{Z}[\sqrt{-2}]$ = $\mathbb{Z}[\sqrt{2}i]$ = $\mathbb{Z}[t]/(t^2 + 2)$ Same as ring generated by character values.
minimal splitting field, i.e., smallest field of realization (characteristic zero)	$\mathbb{Q}(\sqrt{-2})$ = $\mathbb{Q}(\sqrt{2}i)$ = $\mathbb{Q}[t]/(t^2 + 2)$ Same as field generated by character values, because all Schur index values are 1. See minimal splitting field need not be cyclotomic
condition for a field to be a splitting field	The characteristic should not be equal to 2, and the polynomial $t^2 + 2$ must split. For a finite field of size $q$ , this is equivalent to saying that $q \equiv 1 \pmod 8$ or $q \equiv 3 \pmod 8$ .
minimal splitting field (characteristic $p \ne 0, 2$ )	Case $p \equiv 1 \pmod 8$ or $p \equiv 3 \pmod 8$ : prime field $\mathbb{F}_p$ Case $p \equiv 5 \pmod 8$ or $p \equiv 7 \pmod 8$ : Field $\mathbb{F}_{p^2}$ , quadratic extension of prime field.
smallest size splitting field	Field:F3.
degrees of irreducible representations over the rational numbers	1,1,1,1,2,4

Representations

Summary information

Below is summary information on irreducible representations that are absolutely irreducible, i.e., they remain irreducible in any bigger field, and in particular are irreducible in a splitting field. We assume that the characteristic of the field is not 2, except in the last column, where we consider what happens in characteristic 2.

Name of representation type	Number of representations of this type	Degree	Schur index	Criterion for field	Kernel (a normal subgroup of semidihedral group:SD16 -- see subgroup structure of semidihedral group:SD16)	Quotient by kernel (on which it descends to a faithful representation)	Characteristic 2
trivial	1	1	1	any	whole group	trivial group	works
sign representation with kernel $\langle a \rangle$	1	1	1	any	Z8 in SD16: $\langle a \rangle$	cyclic group:Z2	works, same as trivial
sign representation with kernel a maximal dihedral subgroup	1	1	1	any	D8 in SD16: $\langle a^2, x \rangle$	cyclic group:Z2	works, same as trivial
sign representation with kernel a maximal quaternion subgroup	1	1	1	any	Q8 in SD16: $\langle a^2, ax \rangle$	cyclic group:Z2	works, same as trivial
two-dimensional irreducible, not faithful	1	2	1	any	center of semidihedral group:SD16: $\langle a^4 \rangle$	dihedral group:D8	indecomposable but not irreducible
two-dimensional faithful irreducible	2	2	1	The polynomial $t^2 + 2$ must split, i.e., $-2$ must have a square root	trivial subgroup, i.e., it is a faithful linear representation	semidihedral group:SD16	?

Below are representations that are irreducible over a non-splitting field, but split over a splitting field.

Name of representation type	Number of representations of this type	Degree	Criterion for field	What happens over a splitting field?	Kernel	Quotient by kernel (on which it descends to a faithful representation)
four-dimensional faithful irreducible	1	4	The polynomial $t^2 + 2$ must not split, i.e., $-2$ must not have a square root	splits into the two two-dimensional faithful irreducibles.	trivial subgroup, i.e., it is a faithful linear representation	semidihedral group:SD16

Trivial representation

The trivial representation or principal representation (whose character is called the trivial character or principal character) sends all elements of the group to the $1 \times 1$ matrix $(1)$ :

Element	Matrix	Characteristic polynomial	Minimal polynomial	Trace, character value
$e$	$(1)$	$t - 1$	$t - 1$	1
$a$	$(1)$	$t - 1$	$t - 1$	1
$a^2$	$(1)$	$t - 1$	$t - 1$	1
$a^3$	$(1)$	$t - 1$	$t - 1$	1
$a^4$	$(1)$	$t - 1$	$t - 1$	1
$a^5$	$(1)$	$t - 1$	$t - 1$	1
$a^6$	$(1)$	$t - 1$	$t - 1$	1
$a^7$	$(1)$	$t - 1$	$t - 1$	1
$x$	$(1)$	$t - 1$	$t - 1$	1
$ax$	$(1)$	$t - 1$	$t - 1$	1
$a^2x$	$(1)$	$t - 1$	$t - 1$	1
$a^3x$	$(1)$	$t - 1$	$t - 1$	1
$a^4x$	$(1)$	$t - 1$	$t - 1$	1
$a^5x$	$(1)$	$t - 1$	$t - 1$	1
$a^6x$	$(1)$	$t - 1$	$t - 1$	1
$a^7x$	$(1)$	$t - 1$	$t - 1$	1

Sign representation with kernel $\langle a \rangle$

This representation is a one-dimensional representation sending everything in the cyclic subgroup $\langle a \rangle$ (see Z8 in SD16) to $(1)$ and everything outside it to $(-1)$ .

To keep the description short, we club together the cosets rather than having one row per element:

Elements	Matrix	Characteristic polynomial	Minimal polynomial	Trace, character value
$\{ e,a,a^2,a^3,a^4,a^5,a^6,a^7 \}$	$(1)$	$t - 1$	$t - 1$	1
$\{ x,ax,a^2x,a^3x,a^4x,a^5x,a^6x,a^7x \}$	$(-1)$	$t + 1$	$t + 1$	-1

Sign representation with kernel $\langle a^2, x \rangle$

There is a sign representation with kernel $\langle a^2, x\rangle$ which is dihedral group:D8 (see D8 in SD16). Everything inside the subgroup goes to $(1)$ and everything outside the subgroup goes to $(-1)$ .

To keep the descriptions short, we club together the cosets rather than having one row per element:

Elements	Matrix	Characteristic polynomial	Minimal polynomial	Trace, character value
$\{ e,a^2,a^4,a^6,x,a^2x,a^4x,a^6x \}$	$(1)$	$t - 1$	$t - 1$	1
$\{ a,a^3,a^5,a^7,ax,a^3x,a^5x,a^7x \}$	$(-1)$	$t + 1$	$t + 1$	-1

Sign representation with kernel $\langle a^2, ax \rangle$

There is a sign representation with kernel $\langle a^2, ax\rangle$ which is quaternion group (see Q8 in SD16). Everything inside the subgroup goes to $(1)$ and everything outside the subgroup goes to $(-1)$ .

To keep the descriptions short, we club together the cosets rather than having one row per element:

Elements	Matrix	Characteristic polynomial	Minimal polynomial	Trace, character value
$\{ e,a^2,a^4,a^6,ax,a^3x,a^5x,a^7x \}$	$(1)$	$t - 1$	$t - 1$	1
$\{ a,a^3,a^5,a^7,x,a^2x,a^4x,a^6x \}$	$(-1)$	$t + 1$	$t + 1$	-1

Two-dimensional irreducible unfaithful representation

This representation has kernel equal to $\langle a^4 \rangle$ -- center of semidihedral group:SD16. It descends to a faithful irreducible two-dimensional representation of the quotient group, which is isomorphic to dihedral group:D8.

To keep the descriptions short, we club together the cosets rather than having one row per element:

Element	Matrix	Characteristic polynomial	Minimal polynomial	Trace, character value	Determinant
$\{ e, a^4 \}$	$\begin{pmatrix} 1 & 0 \\ 0 & 1 \\\end{pmatrix}$	$(t - 1)^2 = t^2 - 2t + 1$	$t - 1$	2	1
$\{ a, a^5 \}$	$\begin{pmatrix}0 & -1 \\ 1 & 0 \\\end{pmatrix}$	$t^2 + 1$	$t^2 + 1$	0	1
$\{ a^2, a^6 \}$	$\begin{pmatrix} -1 & 0 \\ 0 & -1 \\\end{pmatrix}$	$(t + 1)^2 = t^2 + 2t + 1$	$t + 1$	-2	1
$\{ a^3, a^7 \}$	$\begin{pmatrix}0 & 1 \\ -1 & 0 \\\end{pmatrix}$	$t^2 + 1$	$t^2 + 1$	0	1
$\{ x, a^4x \}$	$\begin{pmatrix}1 & 0 \\ 0 & -1 \\\end{pmatrix}$	$t^2 - 1$	$t^2 - 1$	0	-1
$\{ ax, a^5x \}$	$\begin{pmatrix}0 & 1 \\ 1 & 0 \\\end{pmatrix}$	$t^2 - 1$	$t^2 - 1$	0	-1
$\{ a^2x, a^6x \}$	$\begin{pmatrix}-1 & 0 \\ 0 & 1 \\\end{pmatrix}$	$t^2 - 1$	$t^2 - 1$	0	-1
$\{ a^3x, a^7x \}$	$\begin{pmatrix}0 & -1 \\ -1 & 0 \\\end{pmatrix}$	$t^2 - 1$	$t^2 - 1$	0	-1

Two-dimensional faithful irreducible representations

Four-dimensional faithful irreducible representation over a non-splitting field

Character table

FACTS TO CHECK AGAINST (for characters of irreducible linear representations over a splitting field):
Orthogonality relations: Character orthogonality theorem | Column orthogonality theorem
Separation results (basically says rows independent, columns independent): Splitting implies characters form a basis for space of class functions|Character determines representation in characteristic zero
Numerical facts: Characters are cyclotomic integers | Size-degree-weighted characters are algebraic integers
Character value facts: Irreducible character of degree greater than one takes value zero on some conjugacy class| Conjugacy class of more than average size has character value zero for some irreducible character | Zero-or-scalar lemma

Below is the character table over a splitting field, where $\sqrt{-2}$ stands for any chosen square root of $-2$ :

Representation/conjugacy class representative	$\{ e \}$ (size 1)	$\{ a^4 \}$ (size 1)	$\{ a^2, a^6 \}$ (size 2)	$\{ a, a^3 \}$ (size 2)	$\{ a^5, a^7 \}$ (size 2)	$\{ x, a^2x, a^4x, a^6x \}$ (size 4)	$\{ ax, a^3x, a^5x, a^7x \}$ (size 4)
$\langle a \rangle$ -kernel sign	1	1	1	1	1	-1	-1
$\langle a^2, x \rangle$ -kernel sign	1	1	1	-1	-1	1	-1
$\langle a^2, ax \rangle$ -kernel sign	1	1	1	-1	-1	-1	1
two-dimensional unfaithful, kernel is center	2	2	-2	0	0	0	0
first faithful irreducible representation	2	-2	0	$\sqrt{-2}$	$-\sqrt{-2}$	0	0
second faithful irreducible representation	2	-2	0	$-\sqrt{-2}$	$\sqrt{-2}$	0	0